Data-Driven Fluid Dynamics

• Data-driven Fluid Dynamics is an interdisciplinary field that integrates machine learning, reduced-order modeling, and statistical inference to model complex, multiscale fluid phenomena.
• It uses hybrid models, surrogate operators, and data-driven closure techniques to enhance traditional CFD, achieving significant speed-ups and improved accuracy.
• Practical applications include turbulence modeling, real-time digital twins, and adaptive simulation frameworks that bridge experimental data with numerical solvers.

Data-driven fluid dynamics denotes the discipline in which modeling, simulation, inference, or closure of complex fluid systems is accomplished by statistical learning on data generated from high-fidelity numerical solvers, laboratory experiments, or even direct field measurements. Departing from the traditional paradigm of exclusively first-principles, equation-based numerical simulation, the data-driven approach seeks to learn operators, closures, parametric surrogates, or even surrogate PDEs, enabling efficiency, flexibility, and adaptivity beyond what is possible with conventional computational fluid dynamics (CFD). This field integrates ML, reduced-order modeling (ROM), statistical inference, and hybrid physics-informed strategies to address high-dimensional, nonlinear, multiscale, and multiphysics challenges central to canonical and industrial fluid problems.

1. Taxonomy of Data-driven Fluid Dynamics Methods

Data-driven fluid dynamics encompasses several intertwined methodologies, each distinguished by the manner in which data is used to augment, replace, or constitute the model operator:

(a) Hybrid/Correction Models:

These approaches learn the residual—i.e., the discrepancy—between a low-fidelity physics-based solver (e.g., RANS, LES, or coarse discretization) and high-fidelity reference data (DNS, experiments), and then augment the solver with this correction. The hybridized operator takes the general form:

$$L_\mathrm{NS}[u] \approx f_{\mathrm{phys}}[u] + M(x,u;\theta),$$

where $M(x,u;\theta)$ is the data-driven (often neural network or Gaussian process) correction, trained to minimize the gap with the high-fidelity reference (Iskhakov et al., 2021). This paradigm is exemplified in RANS stress correction, k–ω production term augmentation, and thermal-fluid closures.

(b) Closure Modeling and Data-driven Constitutive Laws:

Here, either part or all of the closure in a reduced or averaged system (e.g., the Reynolds stress tensor in RANS, subgrid-scale fluxes in LES, or two-phase drag in multiphase flows) is constructed directly from data. Models range from pointwise regressors (RF, NN, GP) mapping local invariants or microstructure descriptors to closure terms, to more global or parameterized neural operators (Iskhakov et al., 2021, Metelkin et al., 28 Jul 2025). This approach enables non-intrusive surrogates for closures that elude first-principles modeling.

(c) Reduced-order and Model-discovery Techniques:

Techniques such as Proper Orthogonal Decomposition (POD), Dynamic Mode Decomposition (DMD), and Gaussian Process Regression (GPR) are leveraged to construct low-dimensional representations and learn dynamical evolution in latent spaces (Ortali et al., 2020, Hess et al., 2022, Miyanawala et al., 2018). More advanced schemes, including direct-time neural operators, operator regression frameworks (e.g., mPDE-Net, DeepONet), and Hankel-DMD-based stabilization, have been used to enable global parameterization and long-horizon stability (Meyer et al., 2021, Cheng et al., 2022).

(d) Direct Surrogate/Operator-learning:

Neural networks (convolutional, graph-based, or Fourier operators) are trained to directly map initial and boundary conditions, physical parameters, and time, to full flow-field or physical quantities on structured or unstructured representations (Wang et al., 25 May 2025, Meyer et al., 2021, Agyei-Baah et al., 17 Jan 2026). This encompasses direct-time GNN surrogates, physics-constrained U-Nets for steady-state problems, and GAN-based dynamics-embedding architectures (Rostamijavanani et al., 2024, Wang et al., 2021).

(e) Physics-informed and Scientific ML:

Physics constraints—such as incompressibility, conservation, symmetries, or log-law adherence—are enforced via network design, regularization penalties, or loss function augmentation. Physics-Informed Neural Networks (PINNs), Tensor-Basis Neural Networks, and hybrid scientific ML modules enable frame-invariance, objective consistency, and compliance with known symmetries and boundary conditions (Xue et al., 2024, Lennon et al., 2022).

(f) Nonparametric and Inverse Modeling:

Using data assimilation (e.g., Ensemble Kalman inversion), full-field closure or drag distributions can be inferred nonparametrically from sparse observational or experimental data, forming a "gray-box" inversion framework directly linked to governing equations (Wang et al., 2016).

2. Data-driven Closure and Surrogate Modeling

Closure modeling, surrogate construction, and model reduction form the backbone of practical data-driven fluid dynamics, making high-dimensional, multiscale, or expensive simulations tractable.

POD-GPR Reduced-Order Models:

POD yields an orthonormal basis $\{\phi_i(x)\}$ from snapshot data. A parametric state $u(x;\mu)$ is approximated as:

$u_r(x;\mu) = \sum_{i=1}^r a_i(\mu) \phi_i(x)$

with GPR learning the mapping $\mu \mapsto a_i(\mu)$. This yields speed-ups of $10^2$–$10^4\times$ versus full CFD, sub-percent errors on canonical flows, and enables many-query tasks such as optimization and uncertainty quantification (Ortali et al., 2020).

Hybrid POD–CNN and Neural Operator Models:

Projecting high-fidelity snapshot data onto low-dimensional POD coordinates, temporal evolution is encoded by shallow CNNs or deep surrogates. For fluid-structure interaction and unsteady turbulent flows, CNNs train to predict the progression of POD coefficients, delivering $O(10^2)$–$O(10^3)$ speedups and minimal error accumulation on long trajectories (Miyanawala et al., 2018).

End-to-end Neural Operator Learning:

Feedforward NNs, GNNs, and U-Nets are trained to directly map problem specification to flow fields, either stepwise (iterative forecast) or via "direct-time" predictions. The direct-time GNN surrogate, for example, maps $(t, \lambda)$ directly to the mesh-resolved fluid state, eliminating per-step error accumulation and supporting general unstructured grids (Meyer et al., 2021).

Physics-informed Wall Models and Multiphase Closures:

PINNs trained on high-fidelity LES or IDDES data, under log-law and continuity constraints, predict wall-shear and near-wall velocity in LBM-based turbulence simulations, with below 4% error up to $Re_\tau = 10^6$ and three-orders-of-magnitude reduction in required data versus DNS-based models (Xue et al., 2024). For particle-laden flows, local volume-fraction representations coupled to fully connected NNs achieve $R^2\geq0.92$ on drag, outperforming position-based architectures and offering natural extensions to polydispersity and complex wall effects (Metelkin et al., 28 Jul 2025).

GAN and Generative Frameworks:

Conditional GANs and their extensions (Dyn-cGAN) embed low-dimensional, parameter-dependent dynamics blocks inside adversarial pipelines to generate flow field sequences across varying Reynolds numbers, accurately predicting both velocity/vorticity snapshots and time-series dynamics, with controlled generalization up to $Re \sim 3000$ (Rostamijavanani et al., 2024).

3. Scientific ML and Physics-informed Learning

Incorporation of first-principles physics—be it through hard constraints, invariant architectures, or hybrid loss functions—distinguishes advanced data-driven fluid dynamics from black-box statistical modeling:

Frameworks for Complex Rheology:

Tensor-basis neural networks (TBNNs) and scientific ML models (RUDEs) learn materially objective, frame-invariant constitutive relations from sparse or protocol-heterogeneous data, yielding stress closures compatible with CFD solvers. For example, learned Maxwell–Oldroyd tensors for PEG hydrogels are embedded in OpenFOAM simulations, achieving sub-1% velocity error in 2D contraction flows and robust transfer across measurements and flow regimes (Lennon et al., 2022).

Inpainting and Data Completion:

Two-stage VQ-VAEs, equipped with discrete latent codebooks and GAN plus perceptual loss, inpaint masked turbulent vorticity fields, surpassing FNO and transformer architectures in pointwise, spectral, and statistical accuracy. Discrete codebook enforcement constrains reconstructions to physically plausible patches, stabilizing multi-scale completion under severe occlusion (Shu et al., 2024).

Equation-informed Data-driven Budget Identification:

Sparse regression techniques (SINDy) identify local flow-regime partitions by extracting dominant PDE budget terms per spatial or Lagrangian point, followed by community clustering to reveal dynamically distinct flow regions. This enables automated domain segmentation for hybrid solver switching, region-specific closure adaptation, and interpretability, with proven accuracy in canonical vortex and stratified turbulence flows (Sevryugina et al., 2024).

4. Multiscale, Bifurcation, and Operator Inference

Multiscale Bridging and Hybrid Models:

Data-driven methods bridge micro–macro gaps via error-correction, hybrid closures, and concurrent coupling (e.g., HMM, EFM). Machine-learned surrogates replace or accelerate expensive microscale solvers, facilitate error-correction in system codes, and enable data-informed interface matching, with reported accuracy gains and runtime reductions (up to $100\times$) depending on the approach (Iskhakov et al., 2021).

Bifurcation-aware and Model-discovery Pipelines:

Hankel-DMD overcomes limitations of standard DMD by embedding time-delay, stabilizing predictions through bifurcations (e.g., Hopf, pitchfork), and projecting onto dynamically consistent submanifolds. Localized projection bases, clustered and selected via neural nets, enable accurate tracking of multiple solution branches in parameter regimes with symmetry-breaking or time-periodicity, outperforming global bases by an order of magnitude in error (Hess et al., 2022).

Data-driven Fluid PDE Discovery (mPDE-Net):

Regression over operator libraries, with regularization and sparsity constraints, discovers multi-moment fluid PDEs (e.g., heat-flux closure in Landau damping) from kinetic data, achieving $\leq 0.5\%$ error versus kinetic theory and up to $100\times$ reduction in computational cost (Cheng et al., 2022).

Data-driven Direct Modelling:

New frameworks learn solutions by combining only first-principles constraints (such as incompressibility, balance laws) with discrete experimental data, introducing the concept of model-free, data-constrained weak solutions. $\Gamma$-convergence analyses guarantee that as the data set densifies, solutions relax to the classical PDE limit (if the data reflect a monotone constitutive relation), underpinning a mathematically rigorous route to data-driven PDE solvers (Lienstromberg et al., 2022).

5. Benchmarks, Societal Infrastructure, and Practical Impact

Standardization and Benchmarking (FD-Bench):

FD-Bench establishes the first comprehensive and modular benchmark framework for data-driven fluid simulation, featuring standardized datasets covering a spectrum of canonical and engineering relevant flows. It decomposes models into spatial, temporal, and loss modules, enabling granular attribution of performance gains. Empirical studies show data-driven solvers, such as Fourier Neural Operators (FNO), outperform traditional solvers in both accuracy and wall-clock cost across Reynolds and Mach regimes, recover spectral energy, and transfer across spatial resolutions (Wang et al., 25 May 2025). Leaderboards on representative flows encapsulate RMSE, normalized RMSE, efficiency, and generalization capacity, with code and APIs ensuring reproducibility and extensibility.

Speed, Scalability, and Real-time Applications:

Weakly supervised, physics-informed, and generative surrogate models achieve millisecond-scale predictions for steady-state and dynamic flows (e.g., sub-10 ms for $64 \times 64$ Navier–Stokes on desktop CPU (Wang et al., 2021); up to $28\times$ speed-up for pressure solve in MPMNet (Li et al., 2023)), supporting real-time digital twins, design optimization, and interactive analysis on commodity hardware.

Limitations and Ongoing Challenges:

Despite their promise, data-driven fluid models face notable limitations: generalization beyond the training regime remains nontrivial (especially at high Reynolds number, in highly turbulent, or rare event regimes), and extrapolation to out-of-sample geometries, microstructures, or boundary conditions can induce failure modes (Rostamijavanani et al., 2024, Xue et al., 2024). Embedding strong physical constraints and hybridizing with physics-based solvers is often essential for stability and interpretability. Scalability of certain architectures (e.g., GPR, or large memory CNNs at high resolution) and the need for high-fidelity snapshot data continue to be practical bottlenecks in large-scale deployment.

6. Future Directions and Theoretical Advances

Hybrid Physics-ML Integration:

Hard-encoding of symmetry, frame-invariance, conservation, and realizability in neural operators—via equivariant architectures, loss engineering, or hybrid optimization—remains an open and fertile ground. Examples include TBNNs for complex rheology (Lennon et al., 2022) and log-law PINNs for wall turbulence (Xue et al., 2024).

Operator Learning across Multiphysics and Scales:

Future pipelines will leverage multi-fidelity data, progressive fine-tuning, and adaptive hybrid schemes, enabling seamless transitions between regimes (e.g., RANS–LES, single- to multiphase, or Newtonian to complex fluid transitions) (Iskhakov et al., 2021). Operator regression strategies (mPDE-Net, DeepONet), and data-driven closure in multiphase or particulate systems (Metelkin et al., 28 Jul 2025), will facilitate highly adaptive and domain-generalizable solvers.

Data-driven Inverse and Gray-box Approaches:

Recent advances in equation-informed clustering, nonparametric inversion, and Bayesian assimilation will propagate to next-generation CFD workflows, enabling data-driven calibration, uncertainty quantification, and sensor placement for monitoring, control, and digital twin deployment (Sevryugina et al., 2024, Wang et al., 2016).

Interpretability, White-boxing, and Physical Diagnostics:

Transparent models (e.g., finite-difference CNNs with interpretable weights (Agyei-Baah et al., 17 Jan 2026)), dynamically adaptive clustering of physical regimes (via SINDy or regression-based classification), and sparsity-enforced model discovery schemes, are expected to make data-driven fluid dynamics more amenable to standard scientific analysis and diagnostic scrutiny.

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Data-driven fluid dynamics has synthesized elements of machine learning, model reduction, and physical modeling into a robust and expanding toolkit, catalyzing advances in high-fidelity simulation, accelerated design optimization, data completion, and adaptive closure modeling for turbulent, multiphase, and complex fluids. The field will continue to advance through foundational developments in modular benchmarking, hybrid integration of physics and learning, and the increasing accessibility of both high-resolution datasets and efficient operator-learning architectures.